Problem: Is ${200284}$ divisible by $9$ ?
A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {200284}= &&{2}\cdot100000+ \\&&{0}\cdot10000+ \\&&{0}\cdot1000+ \\&&{2}\cdot100+ \\&&{8}\cdot10+ \\&&{4}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {200284}= &&{2}(99999+1)+ \\&&{0}(9999+1)+ \\&&{0}(999+1)+ \\&&{2}(99+1)+ \\&&{8}(9+1)+ \\&&{4} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {200284}= &&\gray{2\cdot99999}+ \\&&\gray{0\cdot9999}+ \\&&\gray{0\cdot999}+ \\&&\gray{2\cdot99}+ \\&&\gray{8\cdot9}+ \\&& {2}+{0}+{0}+{2}+{8}+{4} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${200284}$ is divisible by $9$ if ${ 2}+{0}+{0}+{2}+{8}+{4}$ is divisible by $9$ Add the digits of ${200284}$ $ {2}+{0}+{0}+{2}+{8}+{4} = {16} $ If ${16}$ is divisible by $9$ , then ${200284}$ must also be divisible by $9$ ${16}$ is not divisible by $9$, therefore ${200284}$ must not be divisible by $9$.